Assimilating Spatially Dense Data for Subsurface - - PowerPoint PPT Presentation

Assimilating Spatially Dense Data for Subsurface Applications—Balancing Information and Degrees of Freedom

Trond Mannseth and Kristian Fossum Uni Research CIPR

Two Porous Media

with different fluid conductivity (permeability) Sandstone sample Sponge

Two Porous Media

with different fluid conductivity (permeability) Sandstone sample Sponge

Task: estimate permeability, k(x)

Seismic Data

Offshore seismic data acquisition

Seismic Data

Offshore seismic data acquisition

Seismic data are spatially dense

Seismic Data

Offshore seismic data acquisition

Seismic data are spatially dense Link between seismic data and k(x)?

Link between Seismic Data and k(x)

Link between Seismic Data and k(x)

k(x) → flow modeling → fluid pressure and fluid content

Link between Seismic Data and k(x)

k(x) → flow modeling → fluid pressure and fluid content

• F. pressure and f. content → petro-elastic modeling → elastic properties

Link between Seismic Data and k(x)

k(x) → flow modeling → fluid pressure and fluid content

• F. pressure and f. content → petro-elastic modeling → elastic properties

Elastic properties → seismic modeling → simulated seismic data

Link between Seismic Data and k(x)

k(x) → flow modeling → fluid pressure and fluid content

• F. pressure and f. content → petro-elastic modeling → elastic properties

Elastic properties → seismic modeling → simulated seismic data We use elastic properties (‘inverted seismic data’) as ‘seismic data’ when estimating k(x)

Background

Inverted seismic data

Background

Inverted seismic data

Elastic properties: Vp, Vs, ρ, . . . are pixel fields

Background

Inverted seismic data

Elastic properties: Vp, Vs, ρ, . . . are pixel fields Spatially dense → high potential for estimating k(x)

Background

Inverted seismic data

Elastic properties: Vp, Vs, ρ, . . . are pixel fields Spatially dense → high potential for estimating k(x) Signal masked by errors (acquisition, processing, inversion, . . . )

Background

Inverted seismic data

Elastic properties: Vp, Vs, ρ, . . . are pixel fields Spatially dense → high potential for estimating k(x) Signal masked by errors (acquisition, processing, inversion, . . . ) ⇒ Extract data features with enhanced ‘signal-to-noise ratio’

Background

Inverted seismic data

Elastic properties: Vp, Vs, ρ, . . . are pixel fields Spatially dense → high potential for estimating k(x) Signal masked by errors (acquisition, processing, inversion, . . . ) ⇒ Extract data features with enhanced ‘signal-to-noise ratio’ Some information will, however, be lost

Background

Ensemble-based methods

Background

Ensemble-based methods

Degrees of freedom (DOF) is limited by ensemble size, E (assuming no localization)

Background

Ensemble-based methods

Degrees of freedom (DOF) is limited by ensemble size, E (assuming no localization) E is usually moderatly large (O(100))

Background

Ensemble-based methods

Degrees of freedom (DOF) is limited by ensemble size, E (assuming no localization) E is usually moderatly large (O(100)) Spatially dense data may lead to unwarranted strong uncertainty reduction in estimation results

Background

Ensemble-based methods

Degrees of freedom (DOF) is limited by ensemble size, E (assuming no localization) E is usually moderatly large (O(100)) Spatially dense data may lead to unwarranted strong uncertainty reduction in estimation results Feature extraction may alleviate this problem

Background

Ensemble-based methods

Degrees of freedom (DOF) is limited by ensemble size, E (assuming no localization) E is usually moderatly large (O(100)) Spatially dense data may lead to unwarranted strong uncertainty reduction in estimation results Feature extraction may alleviate this problem Subspace pseudo inversion is another alternative

Background

Balancing information and DOF

Background

Balancing information and DOF

Both feature extraction and subspace pseudo inversion reduce data influence

Background

Balancing information and DOF

Both feature extraction and subspace pseudo inversion reduce data influence Hence, some of the information available is not applied in the data assimilation

Background

Balancing information and DOF

Both feature extraction and subspace pseudo inversion reduce data influence Hence, some of the information available is not applied in the data assimilation Need to balance the applied information content against available DOF

Scope

Balancing information and DOF

Scope

Balancing information and DOF

How to reduce data influence sufficiently to avoid unwarranted strong uncertainty reduction without discarding important information?

Scope

Balancing information and DOF

How to reduce data influence sufficiently to avoid unwarranted strong uncertainty reduction without discarding important information? Alternatively: How to increase ensemble size sufficiently to handle spatially dense data without increasing computational cost?

Reduce Data Influence

Approaches

Reduce Data Influence

Approaches

Data coarsening

Reduce Data Influence

Approaches

Data coarsening Structure extraction

Reduce Data Influence

Approaches

Data coarsening Structure extraction Subspace pseudo inversion

Reduce Data Influence–Approaches

Data coarsening

2250 2300 2350 2400 2450 2500 2550 2600 2650

Data field 400 data

2280 2320 2360 2400 2440 2480 2520 2560 2600

Reduce Data Influence–Approaches

Data coarsening

2250 2300 2350 2400 2450 2500 2550 2600 2650

Data field 400 data

2280 2320 2360 2400 2440 2480 2520 2560 2600

Coarsened data field 49 data

Reduce Data Influence–Approaches

Structure extraction

2250 2300 2350 2400 2450 2500 2550 2600 2650

Data field 400 data

2250 2300 2350 2400 2450 2500 2550 2600

Reduce Data Influence–Approaches

Structure extraction

2250 2300 2350 2400 2450 2500 2550 2600 2650

Data field 400 data

2250 2300 2350 2400 2450 2500 2550 2600

Smoothened field with 60 data

Reduce Data Influence–Approaches

Structure extraction

2250 2300 2350 2400 2450 2500 2550 2600 2650

Data field 400 data

2250 2300 2350 2400 2450 2500 2550 2600

Smoothened field with 60 data

Structure data: point locations

Reduce Data Influence–Approaches

Subspace psudo inversion1

1Evensen G, Data Assimilation; the Ensemble Kalman Filter

Reduce Data Influence–Approaches

Subspace psudo inversion1

Matrix to be inverted in Kalman gain, W = SST + (E − 1)CD, may be (numerically) singular

1Evensen G, Data Assimilation; the Ensemble Kalman Filter

Reduce Data Influence–Approaches

Subspace psudo inversion1

Matrix to be inverted in Kalman gain, W = SST + (E − 1)CD, may be (numerically) singular Use pseudo inverse, W +, but this is costly for large no. of data

1Evensen G, Data Assimilation; the Ensemble Kalman Filter

Reduce Data Influence–Approaches

Subspace psudo inversion1

Matrix to be inverted in Kalman gain, W = SST + (E − 1)CD, may be (numerically) singular Use pseudo inverse, W +, but this is costly for large no. of data Aproximate W by B = SST + (E − 1)SS+CD(SS+)T, and use B+ in Kalman gain

1Evensen G, Data Assimilation; the Ensemble Kalman Filter

Increase ensemble size without increasing cost

Approach–Upscaled simulations2

2Fossum K and Mannseth T, Coarse-scale data assimilation as a generic alternative

to localization, Comput Geosci 21(1) (2017)

Increase ensemble size without increasing cost

Approach–Upscaled simulations2

Standard forward model: k(x) → f (k(x))

2Fossum K and Mannseth T, Coarse-scale data assimilation as a generic alternative

to localization, Comput Geosci 21(1) (2017)

Increase ensemble size without increasing cost

Approach–Upscaled simulations2

Standard forward model: k(x) → f (k(x)) Upscaled forward model: k(x) → u(k(x)) → f (u(k(x)))

2Fossum K and Mannseth T, Coarse-scale data assimilation as a generic alternative

to localization, Comput Geosci 21(1) (2017)

Increase ensemble size without increasing cost

Approach–Upscaled simulations2

Standard forward model: k(x) → f (k(x)) Upscaled forward model: k(x) → u(k(x)) → f (u(k(x))) Computational cost ∼ solving linear system ∼ O(G β); β ∈ (1.25, 1.5)

2Fossum K and Mannseth T, Coarse-scale data assimilation as a generic alternative

to localization, Comput Geosci 21(1) (2017)

Increase ensemble size without increasing cost

Approach–Upscaled simulations2

Standard forward model: k(x) → f (k(x)) Upscaled forward model: k(x) → u(k(x)) → f (u(k(x))) Computational cost ∼ solving linear system ∼ O(G β); β ∈ (1.25, 1.5) Ensemble computational cost ∼ G βE = G β

u Eu ⇒ Eu =

• G

Gu

β E

2Fossum K and Mannseth T, Coarse-scale data assimilation as a generic alternative

to localization, Comput Geosci 21(1) (2017)

Increase ensemble size without increasing cost

Approach–Upscaled simulations2

Standard forward model: k(x) → f (k(x)) Upscaled forward model: k(x) → u(k(x)) → f (u(k(x))) Computational cost ∼ solving linear system ∼ O(G β); β ∈ (1.25, 1.5) Ensemble computational cost ∼ G βE = G β

u Eu ⇒ Eu =

• G

Gu

β E Cost of Eu upscaled simulations equals that of E standard simulations

2Fossum K and Mannseth T, Coarse-scale data assimilation as a generic alternative

to localization, Comput Geosci 21(1) (2017)

Examples

Setup

Examples

Setup

Original data: bulk-velocity (Vp) pixel field

Examples

Setup

Original data: bulk-velocity (Vp) pixel field Notation for labelling plots:

Examples

Setup

Original data: bulk-velocity (Vp) pixel field Notation for labelling plots: True: results obtained with pixel data and E = 4800

Examples

Setup

Original data: bulk-velocity (Vp) pixel field Notation for labelling plots: True: results obtained with pixel data and E = 4800 Estimate: results obtained with any type of data and computational cost corresponding to E = 100 standard simulations

Examples

Pixel data

2250 2300 2350 2400 2450 2500 2550 2600 2650

t = t1

2250 2300 2350 2400 2450 2500 2550 2600 2650

t = t3

2250 2300 2350 2400 2450 2500 2550 2600 2650

t = t2

2250 2300 2350 2400 2450 2500 2550 2600 2650

t = t4

Examples

k(x) estimate with pixel data on 20x20 grid

3 2 1 1 2 3 4

True mean

3 2 1 1 2 3 4 5 6 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

True stdv

0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20

Examples

k(x) estimate with pixel data on 20x20 grid

3 2 1 1 2 3 4

True mean

3 2 1 1 2 3 4 5 6

Estimate mean

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

True stdv

0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20

Estimate stdv

Examples

k(x) estimate with data coarsening to 7x7 grid

3 2 1 1 2 3 4

True mean

4 3 2 1 1 2 3 4

Estimate mean

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

True stdv

0.16 0.24 0.32 0.40 0.48 0.56 0.64 0.72

Estimate stdv

Examples

k(x) estimate with 98% energy subspace pseudo inversion

3 2 1 1 2 3 4

True mean

3 2 1 1 2 3 4 5

Estimate mean

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

True stdv

0.08 0.12 0.16 0.20 0.24 0.28 0.32

Estimate stdv

Examples

k(x) estimate with 10x10 upscaled simulations

3 2 1 1 2 3 4

True mean

6 4 2 2 4 6 8 10

Estimate mean

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

True stdv

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Estimate stdv

Grid Mismatch

Coarser simulation grid

d1 d2 d3 d4

Data-grid detail

s

Simulation-grid detail

Grid Mismatch

Coarser simulation grid

d1 d2 d3 d4

Data-grid detail

s s s s

Simulation-grid detail after downscaling to data grid

Grid Mismatch

Coarser simulation grid

d1 d2 d3 d4

Data-grid detail

s s s s

Simulation-grid detail after downscaling to data grid

s cannot match four different values

Grid Mismatch

Coarser simulation grid

d1 d2 d3 d4

Data-grid detail

s s s s

Simulation-grid detail after downscaling to data grid

s cannot match four different values Not a problem with upscaled simulations and well data

Examples

k(x) estimate with 10x10 upscaled simulations and 10x10 data coarsening

3 2 1 1 2 3 4

True mean

3.2 2.4 1.6 0.8 0.0 0.8 1.6 2.4 3.2

Estimate mean

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

True stdv

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Estimate stdv

Summary

Summary

Investigated how to balance information against available DOF

Summary

Investigated how to balance information against available DOF Three ways of reduction of data-space influence (data coarsening, structure extraction, subspace pseudo inversion) and one way of increasing ensemble size without increasing cost (upscaled simulations) have been considered

Summary

Investigated how to balance information against available DOF Three ways of reduction of data-space influence (data coarsening, structure extraction, subspace pseudo inversion) and one way of increasing ensemble size without increasing cost (upscaled simulations) have been considered Reduction of data-space influence (all three ways) gave some improvement, with structure extraction as the least successful

Summary

Investigated how to balance information against available DOF Three ways of reduction of data-space influence (data coarsening, structure extraction, subspace pseudo inversion) and one way of increasing ensemble size without increasing cost (upscaled simulations) have been considered Reduction of data-space influence (all three ways) gave some improvement, with structure extraction as the least successful Upscaled simulations did not give good results

Summary

Investigated how to balance information against available DOF Three ways of reduction of data-space influence (data coarsening, structure extraction, subspace pseudo inversion) and one way of increasing ensemble size without increasing cost (upscaled simulations) have been considered Reduction of data-space influence (all three ways) gave some improvement, with structure extraction as the least successful Upscaled simulations did not give good results Upscaled simulations combined with data coarsening gave good results, particularly when scales of simulation grid and data grid were similar

Acknowledgements

Partial financial support was provided by the CIPR/IRIS cooperative project ‘4D Seismic History Matching’, which is funded by industry partners Eni, Petrobras and Total, as well as the Research Council of Norway (PETROMAKS II)